Tricky Fourier Transform problem for an exponential function

[eaglemike]

Homework Statement

find the Fourier transform, using the definition of the Fourier transform $\widehat{f}$($\nu$)=∫$^{∞}_{-∞}$f(t)e$^{-2 \pi i \nu t}$dt, of the function f(t)=2 $\pi$t$^{2}$e$^{- \pi t^{2}}$

Homework Equations

I have the answer:

(1-2${\pi \nu^{2}}$)e$^{- \pi \nu^{2}}$

The Attempt at a Solution

After inserting f(t) into the equation for the transform, I added the exponents on the e terms, factored out -$\pi$, and added and subtracted $\nu^{2}$ to get

∫$^{∞}_{-∞}$2$\pi$t$^{2}$e$^{- \pi (t^{2}+2 i \nu t + \nu^{2} - \nu^{2})}$dt

I then substituted x=t+i$\nu$ to get

∫$^{∞}_{-∞}$2$\pi$(t+i$\nu$)$^{2}$e$^{- \pi (x^{2}+ \nu^{2} )}$dt

at this point I know that I should be able to solve the integral, probably by integrating by parts, but I am really just lost. This seems like such a tricky integral. I thought maybe squaring the equation to get a double integral and then converting to polar coordinates would work but I couldn't get that to work out either. Thanks for the help!

 

[Ray Vickson]

Homework Statement

find the Fourier transform, using the definition of the Fourier transform $\widehat{f}$($\nu$)=∫$^{∞}_{-∞}$f(t)e$^{-2 \pi i \nu t}$dt, of the function f(t)=2 $\pi$t$^{2}$e$^{- \pi t^{2}}$

Homework Equations

I have the answer:

(1-2${\pi \nu^{2}}$)e$^{- \pi \nu^{2}}$

The Attempt at a Solution

After inserting f(t) into the equation for the transform, I added the exponents on the e terms, factored out -$\pi$, and added and subtracted $\nu^{2}$ to get

∫$^{∞}_{-∞}$2$\pi$t$^{2}$e$^{- \pi (t^{2}+2 i \nu t + \nu^{2} - \nu^{2})}$dt

I then substituted x=t+i$\nu$ to get

∫$^{∞}_{-∞}$2$\pi$(t+i$\nu$)$^{2}$e$^{- \pi (x^{2}+ \nu^{2} )}$dt

at this point I know that I should be able to solve the integral, probably by integrating by parts, but I am really just lost. This seems like such a tricky integral. I thought maybe squaring the equation to get a double integral and then converting to polar coordinates would work but I couldn't get that to work out either. Thanks for the help!

Your last expression is wrong: it should be $\int_{-\infty + i \nu}^{\infty + i \nu} 2 \pi x^2 e^{-\pi (x^2 + \nu^2)}\, dx ,$, which can be replaced by the same integral from x = -infinity to +infinity along the real x-axis (why)? Now int x^2*exp(-x^2) dx can be attacked using integration by parts.

RGV